SURVEY ON MODERN RADARSIGNAL PROCESSING

Joao Paulo Goncalves Barcia

OUOLEY KNOX LIBRARYmaval POSTQRAuUATE SCHOOLMONTEREY. CALIFORNIAWMO

b t ii 1/ ri I UQ I U Hi till Oni:f.

vjonierey, baiiTorr

TSURVEY ON MODERN RADAR SIGNAL PROCESSING

by

Joao Paulo Goncalves Barcia

December 1975

Thesis Advisor: John Bouldry— Mi i i i i hi ni'imiii

i mil ii— i ——a—— i ii mil i»i wii in "-—i-- gnnii-wwi 1 <» 1

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Survey on Modern Radar SignalProcessing

5. TYPE OF REPORT * PERIOD COVEREDMaster's Thesis;December 1975

6. PERFORMING ORG. REPOHT NUMBER

7. AuTHOR(»)

Joao Paulo Goncalves Barcia

8. CONTRACT OR GRANT HUMBEF<("t;

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Naval Postgraduate SchoolMonterey, California 93940

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Naval Postgraduate SchoolMonterey, California 95940

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Naval Postgraduate SchoolMonterey, California 93940

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RadarDigitalSignal processing

20. ABSTRACT (Continue on rrveiee tide if neceeeery and Identity by block number)

The purpose of this thesis is to investigate the state ofthe art of radar signal design as well as radar signal processorsand determine the actual trends in modern radar design. The useof a digital general purpose radar signal processor is discussed.The concepts of ambiguity and autocorrelation function areinvestigated in regard to radar resolution capabilities. Theconcept and analytical development of the DFT/FFT is presented.

DDI JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE

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Quantization noise in a digital MTI processor and its effectsin the improvement factor are analyzed. Optimization techniquesfor the response curve of digital MTI processors using staggeredPRF are investigated. The SAR concept and analysis as well astechniques to obtain low correlator rates in SAR digitalprocessors are presented.

DD Form 1473, 1 Jan 7?

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Survey on Modern Radar Signal Processing

by

Joao Paulo Goncalves BarciaFirst Lieutenant, Portuguese Navy

B.S., Naval Postgraduate School, 1974

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAA/AT DnCTrnAniiA-rr nmi.

c '

DUDLEY KNOX UBRAR*NAVAL POSTGRAuUATE SCHOOLHONTEREY. CALIFORNIA 93940

ABSTRACT

The purpose of this thesis is to investigate the state

of the art of radar signal design as well as radar signal

processors and determine the actual trends in modern radar

design. The use of a digital general purpose radar signal

processor is discussed. The concepts of ambiguity and auto-

correlation function are investigated in regard to radar

resolution capabilities. The concept and analytical devel-

opment of the DFT/FFT are presented. Quantization noise in

a digital MTI processor and its effects in the improvement

factor are analyzed. Optimization techniques for the response

curve of digital MTI processors using staggered PRF are in-

vestigated. The SAR concept and analysis as well as techniques

to obtain low correlator rates in the SAR digital processors

are presented.

TABLE OF CONTENTS

I. INTRODUCTION 8

II. RADAR. HISTORY AND APPLICATIONS 10

III. SIGNAL PROCESSING IN RADAR 12

A. INTRODUCTION TO THE RADAR RANGE EQUATION 12

B. RELATION BETWEEN THE RADAR RANGE EQUATIONAND SIGNAL PROCESSING 14

C. USE OF COMPUTERS AS SIGNAL PROCESSORS IN RADAR--- 15

IV. VARIOUS ASPECTS OF THE THEORY OF RADARSIGNAL PROCESSING 20

A. SIGNAL PROCESSING 20

B. AMBIGUITY FUNCTION AND AUTOCORRELATION 23

1. Concept 23

2. Range Ambiguity Function 23

3. The Velocity Ambiguity Function 26

C. PULSE COMPRESSION 30

1. Concept 30

2. Linear FM Chirp Pulse Compression 51

3. Matched- Filter Approach 36

4. Use of Discrete Frequency Sequencesin Pulse Compression 37

D. MOVING TARGET INDICATOR (MTI) 45

E. DIGITAL SIGNAL PROCESSING, DFT AND FFT 59

F. SYNTHETIC APERTURE RADAR 73

V. RECENT DEVELOPMENTS IN TWO MAJOR AREAS 86

A. DIGITAL MTI 86

B. DIGITAL SAR -109

VI. CONCLUSIONS. TRENDS 123

BIBLIOGRAPHY 124

INITIAL DISTRIBUTION LIST 126

ACKNOWLEDGEMENT

The author wishes to acknowledge the support and

encouragement provided him by Associate Professor John M.

Bouldry. In addition, the author wishes to thank all the

Dudley Knox Library staff.

I. INTRODUCTION

The processing of signals has been always a necessity

whenever information was to be transmitted through a channel

between a source and a destination. It was not until 1948

that Shannon presented the basic theory of information re-

lating the source entropy with channel capacity and the

probability of error. Techniques of signal processing were

already in use, but the results given by the information

theory clarified the limits that could be achieved.

Early analog processors with enough bandwidth and reason-

able signal to noise ratio could process data at extremely

high data rates, but the components' low stability and high

cost, and sometimes low versatility, were responsible for a

gradual substitution of analog by digital processors when

the price of digital logic went down, and the digital hard-

ware technology suffered a big jump in the last ten years.

In fact, in today's radar signal processing the trend to

a substitution of analog by digital processors is inevitable.

The theory and boundaries of digital signal processing are

continually being improved as well as hardware components.

Speed is almost no longer a problem and versatility is immense.

Chapter II is an introduction to radar history. In Chapter

III signal processing is related with the radar problem through

the radar range equation and a general digital processor for

a radar system is presented. In Chapter IV an analysis is

8

made of the present signal processing techniques in radar

with special emphasis into digital approaches. In Chapter V

an investigation is made into the recent developments and

problems of two major areas of radar digital signal process-

ing. In Chapter V an overall view of the achievements and

trends of radar signal processing is explored.

II. RADAR. HISTORY AND APPLICATIONS

The electromagnet ism and electromagnetic wave propagation

theories of Maxwell, later (1886) experimentally verified by

Hertz, contain all the necessary background to understand the

principles of radar. The first experiments on detection with

radio waves started in 1903. Due to inadequate technology

the results were very poor. The obtained ranges were less

than those achieved with optical systems and this was enough

to discourage any interest in pursuing the experiments. It

Avas only in 1922 at the Naval Research Laboratory that a

wooden ship was detected using a CW radar with the transmitter

and the receiver as separate units. From then on an increas-

ing interest in radar technology became evident. In 1930

using a bistatic radar, the first aircraft was detected, and

by 1932 the detection ranges were already in the 50-mile

region. By that time the pulse techniques were not yet

explored, so the information was in the presence or absence

of a target and not in range information. But in 1935 the

British successfully used pulse techniques to measure distance

to the targets. In 1936 the detection range was already in

the 90-miles region. The basic principles were understood.

Its next steps were in technology improvements in order to

get stable elements and more power output to increase the

range coverage. Higher frequencies were used in order to

work with smaller elements and high gain antennas.

10

The discovery of the magnetron was an important step since

powers went up by a factor of 100, and wavelengths of 10 cm

were obtained. But the greatest incentive for further devel-

opments in the radar field came from the military necessities

of World War II. After the War a stagnation of about five

years slowed down the rhythm of improvements in the field.

But in the fifties the introduction of the high power klystron

increased not only the power output but the frequency stability

necessary for coherent detection as well as coherent MTI ; also

very low noise receivers were implemented.

From then on, with more and more perfect technology, more

complex systems appeared. The introduction of the computer

as a storage and control element was of extreme importance.

All ballistic systems and very large coverage systems suffered

a big impulse. Digital systems improved the implementation

of stereable array antennas, synthetic aperture radar, track-

ing systems, as well as all digital processing techniques.

With the development of radar, the statistical nature of

the detection problem gave rise to an increasing interest in

the study of the statistical properties of clutter, radar

cross section, rain, etc. So, due to a more precise knowledge

(modeling) of the systems environment, the radar systems

changed depending on the type of application.

Thus, a design to optimize a given application seldom

can be used in others. Today the radar design field is wider.

Applications vary from military to civilian to scientific.

With improvements in technology and more accurate modeling,

research in the field is far from becoming saturated.

11

III. SIGNAL PROCESSING IN RADAR

A. INTRODUCTION TO THE RADAR RANGE EQUATION

Only the analysis of the equation

R = £(ni ,n2

, ... i^) (3-1)

where R is the distance from the radar antenna to the target,

and (n, ,n~, ... nn ) are the radar and environmental parameters,

can give the necessary knowledge to maximize R. If P is the

peak pulse power radiated and G the gain of the antenna, the

transmitted power density is

.

PtG

4 7T R 2

and the received power density, after reflection in a target

of cross section a will be

P G a(3-2)

(4ttR 2)

2

If (3-2) is multiplied by the antenna aperture A , the

received power will be

P. G a. Ap = -1 L_£ (3 _ 3)r

(4ttR 2)

2

If S . is the minimum received signal, then the maximum

distance is

P. G A Q a +it t e t

(4tt) S

r^ = _i E—^_ (3-4)5

min

12

Since by definition of noise figure (F ) of a receiver

S. = N. F ( V^)1 1 n v N ' i = inputo = output

where

Ni

kTBn

input noise

Then, combining (3-4) with (3-5) and (3-6)

R 4

max

PtGAe°t

(4Tr)2 kTB F fS n /N n ) -

*• > n n v o' o^min

(3-5)

(3-6)

Since in a pulse radar the average power is

P = P f xfav t r

x = pulse width

f r = pulse rejection frequency

and the receiver bandwidth B is approximately 1/t

R 4

max

P G A a +av e t

(4i0 2 kTf F (S /N ) .rn u o-/ min

(3-7)

where (S /N ) is the minimum signal to noise ratio at thev o o mm &

output of the linear section of the receiver, necessary for

detection. The first important characteristic in equation

(3-7) is the fact that some of the variables can be considered

deterministic but others must be analyzed as random variables.

The random variables are a. and S/N. Statistical discriptors

for a are already extensively studied [1-2] not only in the

general case but also for some specific cases, as for different

types of clutter. Since the noise is normally treated as

Guassian, all necessary statistical discriptors are available.

13

The statistical discriptors together with the decision

criteria will determine the probability of detection and

probability of false alarm.

B. RELATION BETWEEN THE RADAR RANGE EQUATIONAND SIGNAL PROCESSING

After the War (1945) most of the effort in increasing

range capabilities (eq. #3-7) was focused on increasing

average power (P ) , antenna gains (G) , and mixers and

receivers with lower noise figures (F ) . With the magnetron

and the klystron, the peak powers reached the megawatts range.

Soon it was verified that since the technology was already in

the limits of transmitted peak power, any increase in peak

power was questionable from a technical and a financial aspect,

Also, since

1/4R = kP , k = constant

In R = In k + j In P

d_R = 1 dPR 4 P

That means, that the percent gain in range is only 1/4 of the

percent gain in power. The pulse width could not be increased

much more because of the required range resolution. This led

to the development of what today is called signal processing

techniques. Waveforms were studied in order to increase the

average power but maintaining range resolution. The basic

techniques will be presented in Chapter IV.

The brute force approach to increase the range of detec-

tion (3-7) was used until 1955. The technology then shifted

to an improvement of the signal to noise ratio. If the new

14

signal to noise ratio is

(S/N) = D(S /NQ), D > 1.

Since D > 1, there is an increase in the maximu... range of

detection (3-7)

.

C. USE OF COMPUTERS AS SIGNAL PROCESSORS IN RADAR

As range and aximuth resolution became finer, as multi-

target processing due to digital techniques became more

effective, the conventional final processor in the radar

system, the man, gradually was substituted by the computer.

The total amount of information and the necessary speed for

processing, tax the capabilities of man. So, digital tech-

niques are gradually being substituted for not only those

processors that used analog devices, but also where human

operators were used.

Figure 1 gives an overview of the techniques presently

used in digital signal processing. As in the analog theory

linear approximation is used. So, using the theory of linear

discrete time invariant systems there are two common approaches

digital filtering and spectral analysis. Using the theory of

digital filtering, algorithms can be implemented that correspond

to finite impulse response filters (FIR) or infinite impulse

response filters (IIF). Using spectral analysis, two methods

are possible: implementation of Fast Fourier Transform (FFT)

algorithms or use of statistical spectrum analysis.

Both digital filters and spectral analysis are affected

by quantization noise, which must be taken into account.

15

In

|" ob B

~*

a

Oa

V «**/ aCi<t *~

O »«?

a-

> UJ

< 1</•

>-

1»-

**/>

a */>

S5SIV\ <

£$; _I- <

r

Joz

l. a

2a

zo O </>

** UJ

2 3 JoaBOa u.

IT

16

As seen in Fig. 1, the radar appears as one of the prime

applications for digital systems.

It was in the 1960 ' s that the first low data rate digital

processors were implemented in radar systems, basically to

perform tracking and weapon order computations. The next

step was the implementation of the MTI processor. A flexible

programmable digital processor was not achieved until very

recently. The low cost of semi-conductor memory and the

advanced technology in microprogramming, with inherent increase

in speed, were the main reasons for this achievement. Now a

typical general purpose signal processor for radar will be

described. Figure 2 shows a block diagram of a digital signal

processing system for a radar. The main processing unit is

the arithmetic pipeline, which is a general purpose signal

processor (GPSP) . This consists of five sections selected

in a way that the most frequently necessary operations can

be performed; Fast Fourier Transform (FFT) , recursive and

nonrecursive digital filtering such as MTI, threshold gener-

ation, peak detection noncoherent video integration and range

and angle estimation. The first block in the pipeline, the

matrix switch, selects data from data memories, also from the

receiver through a track buffer, or from anoth er GPSP via a

wrap-around connection. The second block, data scaling,

connects the data for floating point arithmetic. The complex

multiplication block does the equivalent for four real multi-

plications and two real additions, which corresponds to a

complex multiplication. Since some operations, like integration,

17

to

o<

oUJ

\- t- H00 </> 00

_J z z HU — —

6 o e—

e « • eCL 5 ? SH o o o^ n n nOO«;

oQ

•

••

hi

—{^-^

cc

:

u.<

The re-normalization block is sometimes needed. Connected

to the arithmetic pipeline is a memory section and a control

section. The output of the arithmetic pipeline is relayed

to an input/output control unit (IOC) which is a buffer for

the radar data as well as a macro control for the processing

functions of the GPSP. The advantages of this processing system

are that: once the data is started the process is continuous,

and does not follow the normal sequence of fetch, operate, and

store, characteristic of general purpose computers; the pro-

cessor has a horizontal structure as opposed to the normal

vertical structure of computers; this means that an instruc-

tion, after execution, causes all operations specified to be

executed in a clock cycle; there is a hardware separation of

instructions and data in the memory, which simplifies the job

of the programmer; the control memory is made of RAM's which

implies a much greater flexibility in subroutine changes; and

the execution of instructions is within the same time frame

that corresponds to a range or doppler cell.

Four memories interact with the arithmetic pipeline. Two

of them store and recirculate data, the third stores weighting

coefficients for such filters as MTI , FFT or pulse compression,

the fourth stores micro opcodes. Data sequences and arithmetic

operations are both controlled through software, by two control

units. The program that resides in the sequence control memory

(SEQ-CONTMEM) , informs the machine of range and doppler dimen-

sions of the problem. The program in the Arithmetic Control

Memory, ACM, informs the machine of the particular algorithm

to be used.

19

IV. VARIOUS ASPECTS OF THE THEORYOF RADAR SIGNAL PROCESSING

A. SIGNAL PROCESSING

The desire to transmit information is closely related to

the processing of signals. Since normally the information to

be transmitted cannot flow through the available channels, it

is necessary that it be processed to use the chosen channel.

Looking at Fig. 3, the inverse process has to be accomplished

in order to get the information in an understandable form at

the receiving end. The process is not so simple, since in

general the processors themselves and the channel introduce

noise that may or may not eliminate or change the understanding

of the message. So there is a need to interpret the results

with some decision criteria. Here decision theory plays an

important role. Even for the same type of application, for

instance, radar, decision criteria are not the same. The

criteria may be of constant false alarm rate, or may be of

fixed probability of detection. Basically every application

or system must have its decision criteria.

There are two main approaches to the processing of signals,

the time domain approach and the frequency domain approach.

Besides the fact that they use completely different hardware,

in the pure mathematical way, they are duals of each other

through the Fourier transform pair. So the only reason for

one type of approach is due to more perfect reliable type of

hardware

.

20

information processor Processorchannel

1

-

Fig. 3

-

information

21

With the advance of computer techniques , both approaches

are split between digital and analog signal processing.

It was only after World War II that electronic engineers

became interested in the applicability of digital hardware

techniques in signal processing areas. But it was not before

the late sixties that digital techniques suffered a big jump

replacing some of the analog techniques. The foundations of

digital signal processing can be related to the Laplace's

Z- transform theory; but only in the mid sixties was the theory

presented in a formal way. Various papers appeared at that

time related to this subject, but only in 1969 was the first

attempt made for a comprehensive theory of digital signal

processing [21], Most recently (1975), two very comprehensive

books by Oppenheim [3] and Rabiner [4) can be considered as

giving an excellent treatment of the subject, not only through

its mathematical structure but with very good applications.

Today's trend is definitely towards the substitution of analog

processing by digital processing. The speed achieved in

today's digital processors, the very fast algorithms used to

implement the FFT, the gradual substitution of infinite impulse

response methods to finite impulse methods due to a better

knowledge of theory and higher efficiency of calculations are

the major factors responsible for the increasing shift to

digital techniques. There are yet some type of applications

where the speeds are so high, or the digital hardware is so

complicated that analog techniques are still used.

22

B. AMBIGUITY FUNCTION AND AUTOCORRELATION

1. Concept

Considering only non-accelerating targets, the two

kinds of information of interest are the position of the

target and its velocity. Given two targets, the capacity

to differentiate between their positions and velocities is

vital for most radar applications.

The capacity to differentiate in range is called

range resolution, and in velocity, velocity resolution.

As far as range resolution is concerned, it is

obvious the shorter the pulse the better is the differentia-

tion between the two targets, that is, the higher is the

range resolution. With the measurement of velocity, since

it is directly related to the phase difference between sig-

nals, the longer the pulse the greater the number of cycles

that can be compared and consequently the better the doppler

and the velocity resolution. So, at first glance it looks

like to optimize one of the resolutions the other will have

to be sacrificed. The other solution is to try to get an

optimum solution for both cases at the same time. There is

a quantitative way to express these conditions in a precise

way. That was the reason for the appearance of the mathe-

matical concept of ambiguity function.

2

.

Range Ambiguity Function

Let the transmitted signal be represented by S(t) =

Re[\jj(t)], i|'(t) = u(t) exp. j oo t. Then, with no doppler

present the received signals from two different targets are

23

Y(t) and lp(t-x)

t= equivalent time difference between the two targets.

If it is chosen, as a measure of the resolution, the mean square

of the difference between the two received signals, then

+00

e2 (t) = /JiKt) " *(t - x)| 2 dt

+°° +00

= /J*(t)|2 dt + /J,Kt - T)| 2 dt

+<x>

- /0o[^(t)^*(t - T) + l|>*(t)Tp(t - T)]dt

+00

= 2 /Ju(t)|

2 dt - 2Re[exp-j(jj T[/u*(t)u(t - x)dt]]

Since the first part of the result is proportional to the

total energy of the signal and has a constant value, only the

second term is going to make £%t) vary. Therefore

+00

C(T) = / U*(t)u(t - T)dt,* = complex

conjugate

is defined as the range ambiguity function; only c(t) varies

the value of e2 (t). As can be seen, there is a perfect

identity between c(t) and the autocorrelation of the signal.

The ideal case would be to have the ambiguity function with

a spike at the origin and zero anywhere else. That would mean

that the only situation where it was impossible to differen-

tiate the two signals was when they were on top of each other.

As an example. Fig. 4(a) shows the autocorrelation

function of a pulse of duration T, Fig. 4(b) its uncertainty

function |c(x)

|

2. In part (c) there is a rectangle with the

same height of c2 (o) and with the same area under |c(x)| 2

,

24

(a)

-T T

(b)

—AT -

Fie. 4

2 5

that is, A = B. As it can be seen, the base of that rectangle

is a measure of the spread of the curve and is called the

delay resolution constant; so

_ / lc(T)[ 2 dTZiT — -co—

'

'

c 2 (o)

sometimes is more practical to define At as a function of the+00

bandwidth. Let u(w) = F[u(t)] = fm u(t) exp - joit.

since by Parsevall's Theorem F [c(t)] = |u(w)| 2

2tt / |u(co) I MwAt =

[.OuHfdw] 2

if the effective bandwidth is defined as

[/+c°|uU)| 2 dco]

CO 1

We

**iT\uM Pdu

At2 W

e

and the range resolution becomes

AR = ^1-2

All these definitions make sense since as pointed out

before, with a short pulse there is a higher ran- resolution,

but to have a short pulse there is need of a wider bandwidth,

so there is equivalence in the two expressions for At.

3 . The Velocity Ambiguity Function

Using the same type of mathematical logic and defining

*(f) = FU(t)]

26

a complex correlation function is determined [5] to be

K(fd ) = /u*(2irf)u(2Tr£ - 2-nf£df

= /|u(t)| 2 exp j2fr£jtdt

and the doppler resolution constant [5]

A£ = /lK(fd)l

2 dfd = /|u(t) TdtE J_

dk 2 (o) [/|u(t)

|

2 dt] 2 " T e

where T e is the effective duration.

. r 2vf

o

Since ±a =u c

df, = 2dv fo

which infers A f

^

2 f o Av

the velocity resolution constant is

cAv =2 fo Te

The graphical interpretation for T e is similar to the

graphical interpretation given for At in Fig. 4 but instead

of using the function |c(t)|

2,

|K(fj)j

2 is used. But as seen

earlier, the optimization of one resolution implies the mini-

mization of the other. This implies the need to study a two

dimensional correlation function in order to see the mutual

effects between the two resolution parameters.

So if the transmitted signal is

i|j(t) = u(t) exp j2iTfot

the received signal with doppler and delay will be

ip(t-x) = u(t-i) exp j27r(f - fd ) (t-x)

27

The mean square of the difference

£2 = / |iKt) - ij> (t-x)

|

2 dt

will yield a two dimensional correlation function of the form

X(T,fd ) = /u(t) u*(t- T ) exp j27Tfd tdt

Easily it can be seen that

X(t,0) = c(t)

X(0,fd ) = k(fd )

If the volume under |X(T,f,j)| 2 is determined and

divided by |X(0,0)| 2, an equivalent resolution parameter

called the effective area of ambiguity is obtained.

|X(0,0)| 2

It is easy to prove [6] that

//|X (T,f d )|2 dTdfd = |X(0,0)| 2

A(i,fd ) = 1

So if an effort is made to improve one dimension, the other

will never get better. But given one acceptable resolution

in one dimension, there are ways to find the optimal solution

for the other dimension if X(x,f d ) is known. Fig. 5 is an

ambiguity surface for a monochromatic pulse with a smooth

envelope. It would be a reasonable temptation to try to

design the waveform from a desired ambiguity function. That

turns out to be impractical. First, a criteria to design the

ambiguity function would be difficult to build. Secondly, and

28

X { V, j,, )>

Fig. 5

29

most importantly, when designing a radar system normally there

are other more important factors, from economic to space and

weight constraints as well as technical, that dictate the

boundaries for the waveform that we have to pick. Also, there

is an important relation between the chosen waveform and the

type of clutter model, and the approximation of the clutter

model. Normally some flexibility in the digital processing

is desirable.

C. PULSE COMPRESSION

1 . Concept

Soon it was found that trying to increase the maximum

range by increasing power, keeping the same pulse width, was

the difficult way. Isolation problems were difficult to

handle as well as reliable components. The only way to in-

crease the energy of the signal was by increasing the pulse

width. But an increase in pulse width would reduce the range

resolution. Since the real problem was to increase the time

of transmission but at the same time do not decrease the

bandwidth (effective) of the waveform some different types of

waveforms were studied, using the ambiguity function concept.

The processing of the signal should be such that the trans-

mitted waveform with high time of transmission and small

effective bandwidth (We ) should be converted to a high energy

pulse with pulse width approximately 1/We . This is called

pulse compression. There are two basic ways of implementing

pulse compression techniques. The active way, which is a

time domain approach which basically uses correlators in the

30

detection and active elements to produce the waveform in the

transmission. The passive way, which is a frequency domain

approach, uses passive filters to generate the waveform and

matched filters in the receiver. Each of the two has some

variations, and it is even possible to build a system as a

combination of the two processes. Table I gives a relative

performance of various types of pulse compression techniques,

2 . Linear FM (Chirp) Pulses Compression

First, the passive generation of linear FM signals

will be analyzed. An IF pulse generator feeds a dispersive

delay filter with a frequency versus time characteristic as

in Fig. 6(c). The signal is then up converted and only one

of the bands is transmitted. At the receiver the signal is

down converted to IF and then passed through a dispersive

filter with the opposite slope (Fig. £ c) . Using at recep-

tion a mixer sideband inverter, the same type of dispersive

filter can be used because as it can be seen, f~ and f,

(Fig. 6 b) have opposite frequency versus time character-

istics. In fact, with this technique the same dispersive

filter for both transmission and reception can be used.

The analysis approach to this type of processing is

the following:

Let the received signal be of the form

f(t) = exp j [ (a) + u)d ) t + |ut 2], |t| < T/2

= 0, |t| > T/2

where u = ~ = frequency slope of the dispersive filter

31

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32

(A) GENERATOR AND DECODER

MW

f, (t)

i ift)

f 4 (t)

,

<c> FREQUENCY VSTIME

ft WAVEFORMS (ARBITRARY TIME SCALE)

Fig. 6

33

T = duration of the transmit pulse envelope

tod = doppler shift.

The pulse spectrum is

+ 00

F(u>) = f_ m f (t) exp -jut dt

=/«, exp j[(io + wd " w)t +

-^ y t 2] dt

The filter transfer function is

H(w) = exp j [(oo - io)2/2u]

which implies G(oo) = H(w)F(w) = exp j [ (io - co)2/2]j]

T/2x

/ exp j [ (o) + 03^ - o))t + yyt 2] dt

-T/2 *

which implies the time function at the output of the filter

is

-1 1g(t) = F [G(uO] = 27 f ~ GM ex P J wt dw

+00

which implies g(t) = j^- f_ m exp j (to - w) 2 /2y

T/21

/ exp j [ (io + 03d " W ) T + tPt2]

-T/2Z

dx exp jtot dto

Inverting the order of integration:

T / 2 +0°, , fton - to) 2

(t) = / / exp j [(a) + 03d - u))t + 2^ T +2u

+ wt]-T/2 -°°

dwdx

T/21 u 2

= / exp j [oo t + wdT + 7-yT2

+ y?- ] x

-T/2L y

+ OO 9 , .

J" exp j [- cot + £- + tot - -^ o)j do) dx-co ^ ^ L 2u u

34

Call v = (to n + yr - yt)

/T)T

.2 _ WQ2y

1 2 1 2- yxt - w t

Multiplying g(t) by exp -v 2 and exp v 2

1 TV 2 1

+ 0O

Since

,

x fm exp j [to /2y~ v] 2 /2 y doodx

2/2~y v a)[a) - /2~y v] 2_ b^_

2y 2y+ v

2y

~— + v - — a) [ w n + yx - ytl2y y L °

2 ,w .

= V - COT + o— + (Otw

2y(0

If u = CO 2y v

/TU, the second integral will be of the form

+ 0O

2y / exp j u du = /2~y vHT exp j tt/4

which implies g(t) = , exp j (co t - j yt1 ^2 TT-.

+4°

4/ 2exP jT(o)d + VJt) dT

Since co t + wdt + y ^ t2 " y2 =°°dT

+ ^ Tt "2

yt2 +(Jio t

which implies g(t) = /—y- cos O t - T yt 2) sin (a)d + yt)T/2

/(^)

Z(o>d + yt)/2

/ 9 T 2 i

which implies g(t) = (-^ )cos (<o t - y Vt 2) sin x

where x = (co d+ yt)T/2

x2 2

35

sin xFrom these results it can be seen that if w, = 0, aa ' x

type of envelope is obtained with a peak at the origin of

magnitude /r2AojT"T . So the higher AtoT the better the outputIT

signal. If wj / there is a shift in the sin x/x curve which

implies a range error for doppler frequencies greater than

zero

.

3 . Matched-Filter Approach

The matched-filter approach is another passive method.

The analysis of the former will give us a basis for comparison

Let the received signal be

f(t) = cos [(u> t + wd)t + \ Pt2

] , |t| < T/2,0 elsewhere

The output of the matched filter will be of the form

g(x) = /*°°f(t) h (t - t) dt

where h(t) =Kf(-t) and F.[h(t)] = H(w) is the filter transfer

function. K is a constant factor to give a unity gain. In

this case, K = /(-2yu . SoTT

g(T,03d)= /^ZTk ij^os [(w + w d)t + y ^ t2 ]

TT

cos [oj (t - t) - |y (t - T) 2 ]dt

after some steps [7]

(T,<od ) -(Ijl cos[(«n ^)x] sin ("d *^ (T - |t|)]

IT

for I t ! < T

or, g(T..d ) - \ (T- |t!)V^cos[( Wo + ^)t] *i5_*, |xj <T

where x = u2— ( T "

l

T U

36

Since x is not a linear function of t, the shape of

the curve, unlike in the previous case, is not a sin x/x type

of curve. Also, since w^x does not represent a linear shift

in the axis due to the non-linearity of t with respect to x,

if wd / the curves become distorted and not symetric with

respect to the vertical axis as in the previous case (Fig. 7).

The advantage is of course the maximization of the signal to

noise ratio.

4 . Use of Discrete Frequency Sequences in Pulse Compression

The use of a sequence of frequencies each with a pulse

duration of t in order to get a good autocorrelation function

is a very common method.

Various combinations, from linear stepped frequency

to randomly chosen frequencies, are used in order to increase

the main lobe with respect to the side lobes in the ambiguity

function. The basic format is that of Fig. 8

A f,

nr~t •

f. f f.

Fig. 8

where the waveform of the nth segment can be expressed as

vn(t) = A exp j 2ir(£

nt + <j>n )

37

•H

38

The mathematical analysis for the general case is

quite difficult and of little interest. Since only a few

cases are of interest, consider the linear stepped frequency,

that is, a time function of the form,

v(t)N-l

n=0| n[u(t-ni) - u(t-(n + 1)t) ] cos (to + nAw) t

where

Aoj

N

= lowest frequency to be transmitted

= frequency spacing

= number of frequencies in transmission,

Assuming that there is coherency between all frequen-

cies, that is, cj) = 0, which could be obtained using a frequency

synthesizer with a master oscillator, the matched filter would

be of the form indicated in Fig. 9.

nPut I

y r r

t,

A, Ah

r - r

t If

A a AVI

it summer

output

Fig. 9

39

If xAf > 1, there will be significant range ambigui-

ties. Also, if xAf < 1 the subpulse filters f of Pig. 9n b

will overlap. That is the reason why in most cases xAf is

made equal to unity, that is, x = ^. With that assumption,

the convolution of the impulse response of the matched filter

h(t) = v(-t) will yield an output

e (t)

N-l

nl QA* exp j (a)

o+ nAw)t

where (N-l) t < t < Nt .

If A = 1, for all n, thenn '

N-leQ(t) = exp j aj

Qt

n S(exp j Awt)

n

Since | exp j Awt [ < 1, then

eft) = [exp j [w + (N-l)Ao)] t] x sin N (Aoo/2)to

sin (Aw/ 2)

t

Accounting for the autocorrelation of the rectangular

pulse, the final output time function is

e ft) = (1- |x|Af) sin N(Aw/2)tr

(N-ljAio-,

sin (Ao)/2)tC0S LW

o 2J L

Fig. 10 shows the effects of xAf f 1. The nules of

the output waveform occur at sin NAoo/2t = 0.

Which implies N ~f-t = ± m tt

For N > 50, the first side lobe is 13.46db below the main

lobe.

40

•oM

\ . SUBPULSE AUTOCORRELATION

Fig. 10

41

A very important quantitative factor in all these

waveforms with sharp autocorrelation function is the pulse

compression ratio. It is defined as the ratio of peak power

after compression to peak power before compression. Table II

gives some data including compression ratios for some typical

linear pre-pulse generation and compression circuits.

Another important quantitative factor is the side lobe

reduction factor which is defined as the power ratio of the

main lobe with reference to any side lobe.

Depending on the specific application, it is sometimes

preferable to sacrifice the main lobe peak power or even width

(3 db point) , but get a higher side lobe reduction. In the

radar problem those side lobes may be identified with targets,

the need to suppress them if wanting to use a larger dynamic

range is obvious.

There are various types of weighting functions. The

weighting is applied in the amplitude response of the matched

filter in order to deliberately produce a mismatch.

The frequency response of the matched filter is such

that the signal to noise ratio is maximized. So any mismatch

will lower that signal to noise ratio.

The loss in signal to noise ratio due to weighting

is called the loss factor L . It is defined as

rp /xn • ,. , r/rptoCt) dt] 2

i = (S/N) weighte d _L T * J i

Ls (S/N) matched Tf^ z (t) dt

w(t) = weighting function

T = processing time interval or the length of the

transmitted waveform.

42

- ^£o n.O Ojo CX.

co

2 oto ——

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COCO co ro T CN CO CO CN CN c, CN "-

M .g c / Arjo caCU -3

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=»

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co d d o" d d d — d d d d d d —

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d d1— c < < < c <

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.>^

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E •—

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15

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, N ( v x-s ^-^ s-^ ,~-s ,—v^«, <N co ^j- «Tl vO r^n«** ^-^ ^-^ v—' v—

'

Nw^ ^^

H

43

In the case of discrete frequency amplitude weighting, the

loss factor is

N

[I An ]2

Ls

=-hr r 8 J

N[E A2!

The problem of weighting is in effect a way of producing a

pulse compressed spectrum that yields the wanted waveform.

In the continuous linear FM case, the spectrum of the

match filter is of course a pulse in the frequency domain.

Since the inverse transform of a pulse is a sin x/x type of

function, that is why for the linear FM waveform a sin x/x

type of output is obtained. From the Fourier transform pair

it is known that a wide pulse in one domain corresponds to a

sharp pulse in the opposite domain. So the amplitude weight-

ing functions that will lower the- side lobes will have a bell

or tapered shape characteristic in the frequency domain. That

is why the time output of a matched filter with a pulse in the

frequency domain as transfer function, is

1 we„(t) = o— / a(<jo) exp j go t d coo v ' Ztt-w

where a(co) is the weighting function; the most commonly used

weighting functions are:

The cosine function where

a(uO = cos j^ where a(w) = 1 at the center of the

spectrum.

So, e (t) = ^ /cos gexpj utd w =^.^tw* C0S wt '

44

The main lobe will be between nules. The nules will

occur at cos wt =

which implies wt = (2n + 1) tt , n = 0, ±1 ±2

or t = (2n + 1)tt

2w

Considering the lobes between nules the side lobe reduction

is 23. 5 db.

Another class of tapers are the Hamming functions

which have the general form

G O) = a + (1 - a) cos ( -^) < a < 1

where the center of the spectrum is assumed to be at w = 0.

In this case the output will be

^r-o_asinwt 1-ae° UJ ~ 2 wT~ 2w(l-t*)

for a = . 54 the side lobe reduction is 42.8 db which represents

a much larger dynamic range. Table III gives data on side

lobe suppression on frequency coded waveforms.

D. MOVING TARGET INDICATOR (MTI)

The MTI is basically a processor that separates moving

targets from clutter. Since the clutter spectrum is a very

low frequency spectrum, the MTI processor must be a high pass

filter. In the early MTI processors the main technical

problems were in the processor itself clue to instability of

some elements. Today the main problems result due to the

existence of a variety of types of clutter which make the

design difficult. If only the elimination of one type of

45

>>C3<-> *(J r-a 5 m »r

^» ^-*-*^^.^, V, *^ *^, ^. *^ *^*""" "~— -

i 13

c/5

ca"O

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•3C/i

M •

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s- \fl "* O — OOCO CO Ov£>O r-i rr ^j-_ r-_ >o — o sc r-4 ro —_ rs i rr vc

C ou. —

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x> .5O vO O O "frO rj-_ r-; ^r ^r r-

c c •J P—4 —• —* —* «~* •—

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1 ^c ^>— c; ,Jrs o o \c r- Ni/iCinvi-Ov. M"t- O= £ s O in TT mininviv^MriMnf 1 m I

* _; —; ,—' ' —; • '

<

Jurs

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EH

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Tj- O N C Mts in m m m m

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c.2 > >Q o or; "o j~ .2

a 2 ^* xi o ii ii ii n ii

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oo g

5 G

E .E EEE-=-=^5t qCCC — — ^>\n .irsrsrjGCCra

i-H

o „

o -'S c

46

clutter is required, then the efficiency of the MTT is good,

but if various types of clutter must be handled at the same

time by the same processor, the efficiency drops.

Depending on the parameter used, the MTI processors are

basically divided into three types: those that use the phase

information of the returned signal, those that use the ampli-

tude, and those that use both phase and amplitude. The basic

configuration of the phase processing or coherent MTI is that

of Fig. 11. The use of two oscillators, the stalo and coho

,

is for up and down converting the waveforms in order to get

a perfect phase comparison, since at low frequencies the phase

accuracy measurement increases. Coherency is also obtained.

Consider two signal returns from the same moving target

with a separation in time by T = 1/

£

Rwhere f„ = pulse

rejection frequency. The signals presented to the summer are

E-, = E sin (u)dt +<J)-,)

E2

= E sin [ojj(t + T) + 4>2

]

*i=

*z= 4ttVa =

*

The output of the canceller will be

E r= E

x- E

2= 2Esin(^) cos[a>

d(t + |) + <f>]

R

where it is seen that the cosine waveform is modulated by a

sin ^S^- . So, independently of t, there are some doppleri R

frequencies where the power out will be zero. These are the

frequencies where -^ = niT . That is, fd

= n£R(multiples

of the pulse repetition frequency)

.

47

Ef= E(.)-E<t«-T)

Fig. 11

48

Since the doppler frequency is a function of the radial

2Vrvelocity V

R , f, = —— , there are also blind speeds that

correspond to the blind frequencies f £ , = nf„l1 v d R'

Vblind= n (~2K ) >

n is an integer.

Fig. 12 shows the power response for a single delay line

MTI processor. As it is seen, the multiples of the PRF are

at nules but the other frequencies are not processed in the

same way. That is very far away from the ideal case.

The clutter spectrum beating with the pulse repetition

frequency is translated to every multiple of the PRF. So if

the clutter spectrum is, after translated, of the form indi-

cated in Fig. 13 (a) the ideal MTI processor should have a

response as in Fig. 13 (b) .

As it is known the single delay line processor response

is far away from that of Fig. 13 (b) . One way to improve the

MTI processor is by using multiple canceller filters.

For an n-stage cascaded canceller (Fig. 14)

(c~)n = 2 sin (—f ) [9]

b>l XR

where S and S. are the peak output power and peak input power

The problem is that if many cascaded blocks are used, the

Eld-211

(sin —jp4 function may cut some low velocities of interest.

The transmitted waveform has a spectrum as in Fig. 15.

The fact that there is more than one line in the spectrum is

of no importance to the MTI processor since all the lines are

at blind frequencies.

49

o *

4 •'" '"'/'^ r0R OPTIMUM ?

V SI

F0R OTHER VALUES OF f

FREQUENCY RATIO f /»

Fig. 12

50

"\

1

<^2

<=v

3fi.

1 „y 2 3 Vf(b)

/T<

Fig. 13

ZZh -r^-p-

n stages

Fig. 14

51

tJJVELOPE

FREQUENCY

Fig. 15

52

The amplitude processor, also called noncoherent processor,

has a block diagram as in Fig. 16. The advantage of this

processor is due to the fact that stability problems almost

disappear since there is no coherency. Since most high power

transmitters are not very stable, this is a real advantage.

The big disadvantage is that it needs a permanent existence

of a strong clutter return since without it, it won't work.

Fig. 17 represents the phasor diagram that explains how the

noncoherent processor works. Since the clutter is of the

form, E = dc value, in a phasor diagram is represented by a

stationary vector.

Then

Let E (t) be the return of a moving target

Eft) = IE ft) I cos co A ts ^ J' s ^ ;

' d

Es(t + T) = |E

s(t)

|cos [ai

d(t + T)]

Let Wit =<j) - A(J>/2

a) d(t + T) = $ + A(J)/2 = 4> - A<J>/2 + w

dT

which implies WjT = Ac})

Let the total signal at time t (Fig. 17 b) be

^w tc

tsw

and the total signal at time (t + T) (Fig. 17 c) be

E^(t) = E1(t + T) = E^ + E^Ct + T).

From Fig. 17 (d) it can be seen that due to E£

there exists

an amplitude difference between |E1(t)| and |E

2(t)|. This

difference in amplitude is going to be used to identify the

53

t« >TR

mixer

pow.

osc. mua

local

• osc.

delav-E(t)

equal, \-

E(tH)

Fig. 16

54

E/t)=E(t+T)

(b)

(c)

(d)

EA=IEI-IEI

moving target. In order to get better results, instead of

the difference in amplitude, the difference of the square of

the amplitudes will be used. That is the reason for a square

law detector in Fig. 16.

From Fig. 17 (b)

:

Ef = E 2 + E| - 2E 2 E| cos (<j>- Ac|>/2)

(the equation relates only the absolute values of

the vectors)

From Fig . 17 (c)

:

E 2

2= E* - E| - 2E

cEs

cos (4, + A(j>/2)

(only amplitude relations)

Then E 2 = F 2- E 2 = 4E E sin <f> sin ^r 1 2 c s N

2

since (j>= ojjt + A c|> / 2

and Ad> = u ,T = -^J^1

, T = ~d f

R£R

which implies

E 2 = 4EcEs

sin ~^— sin (ojt + A<|)/2)

So, the blind speeds are the same as in the coherent

processor, and the transfer function must have the same shape.

But if E =0, that implies E =0. That is the reason why

an amplitude processor must have a clutter return, and the

stronger the clutter return the better since E increases& r

linearly with E .J c

Clutter cancellation can also be made at IF frequencies

instead of at the video part of the receiver (Fig. 18).

56

(jcCD

oc-

"aOLiJb

1

°U OC03 1 "aU >

lT) h-j

a

^lJ

N

CO

t>0

•H

1

—

-E,

^

-i->

D*-ac

57

Let the two IF signals separated by T = i- befR

E1

= E sin [2ir(flp

± fd) t -

<P q ]

E2

= E sin [27r(£IF

± £d ) (t + T) -

<P Q )

where 6 =4tt£

1FRo

o —

Er

= El

" E2

= 2E sin [^ CfIF

-+ fd)T] cos

[2ir(£IF

±£d)(t + ^) - *

Q ]

is the output of the summer.

The output of the phase detector would be

EQ

= E sin [Tr(fIF

±fd)T] cos {2Tr[f

dt + (f

ip± fj) |] - *

q}

where the video envelope is of the form E sin [it (f T „ ± f ,) Tl .

ir d

Since in an MTI processor if f , = the output must be zero

that means that ^fjpT = niT is a necessary condition. That is,

f = =- = nfR , the IF frequency must be a multiple of the

pulse repetition frequency.

The existence of blind speeds can constitute a problem

since every velocity is not processed in the same way. One

of the most often used methods to improve that situation is

with the use of staggered PRF systems. There are some quanti-

tative factors that describe the efficiency of an MTI processor.

Since they are the descriptors of the MTI processor, it is

worthwhile to state their proper definitions.

Improvement factor is defined as I = r / r -;> where rQ

is

the output target to clutter ratio and r. the input target

58

to clutter ratio. This definition reflects the gain as well

as the clutter rejection of the processor.

Subclutter visibility is defined as the capacity of a

radar to detect moving targets in a clutter environment. A

radar with x db of SCV is one that is able to detect a target

over a clutter that has a signal x db stronger than the moving

target. The SCV cannot be used as a parameter for comparison

between radars since the target to clutter ratio is a function

of the size of the radar resolution cell.

The two main problems with MTI processors design are:

first, a correct modeling of the expected type of clutter,

and second, the correct shaping of the transfer function

depending on the type of clutter. In [2] there is an extensive

description of the different types of clutter and modeling

processes

.

When the platform of the radar is moving, the stationary

targets will have a doppler return different from zero. This

is the typical case of the Airborne MTI processors. Various

techniques of clutter looking and automatic tracking must be

used. The techniques get much more complicated and the

processor becomes much more dependent for specific application.

E. DIGITAL SIGNAL PROCESSING, DFT AND FFT

Due to high speed, low cost, versatility and almost inde-

pendence of external condition of recent digital computers,

the processing of radar signals in digital form covers now a

wide spectrum of the radar signal processing.

59

The two basic methods of approach are represented in

block diagram in Fig. 19. In the first case (Fig. 19 a) the

signal is first sampled and quantized in such a way that the

signal is converted into a sequence of numbers. The numbers

are then stored and the processor, using arithmatic and logical

operations, manipulates the numbers according to an algorithm

which is a function of the type of filter we want. The

resultant numbers are then dequantized and passed through a

low pass filter (D/A) . The second method (Fig. 19 b) is

basically a frequency domain approach of the problem. The

signal is quantized and a discrete Fourier transform algorithm

is applied to the quantized signal. The spectrum of the signal

is then weighted as a function of the type of filtering re-

quired. Then, the inverse operations take place. Due to

recent improvements in faster algorithms to implement the DFT

this second method is becoming widely used. The theory behind

the first method is based on the Z transform theory [10]. The

variable Z represents a delay and is defined through its Laplace

transform by

Z = exp ST

where 1/T is the sampling frequency. Given an analog function

f(t), the transform of f(t) will be

F(Z) =nfQ

f(nT)Z"n

The two basic configurations are recursive and nonrecursive

filters. In the nonrecursive case the output is of the form

Eq

= B^l + A1Z"

1+ . .. A

nZ"

n..] ,

60

oaca o

<

—

^ CD

T3 M-

Q<

ao

1

a

03

en

H

a.ro .zz

61

where the coefficients Ai

are obtained from the coefficients

of a Fourier series expansion of the frequency domain function

of the desired filter.

The feedback or recursive filters are normally obtained

using the analog transfer function of the required filter and

substituting S by a chosen relation between S and Z. Usually

the bilinear transformation

[11] ST -1

1 + ZL

is used since with the correct sampling frequency it is very

accurate and easy to implement. Figure 20 shows the basic

configurations for the nonrecursive (Fig. 20 a) and recursive

(Fig. 20 b ) filters.

In many radar applications the discrete Fourier transform

is being used especially as a bank of filters for obtaining

doppler outputs. The understanding of what it is and its

limitations are thus fundamental.

In order to get a Fourier transform with a digital pro-

cessor not only the input waveform must be in digital form

but also the values of the Fourier transform must be repre-

sented by discrete values. Figure 21 gives us the graphical

derivation of the discrete Fourier transform pair. The input

waveform h(t) (Fig. 21 a) must be sampled in order to be

manipulated by a digital processor. The sampling operation

is the same as multiplying h(t) by a string of delta functions

(Fig. 21 b) . Since multiplication in time is the same as

convolution in the frequency domain, the resulting spectrum

is that of Fig. 21 c) . Here it is seen that if the samples

62

H(z)=-E^1=i +aE(2)

E-

(a)

,z+ - +Az'

)=

E.(z) _ AAz"iE..(2) Z"-B,z"-

(b)

+A„

Fig. 20

63

A (f)

i

I! i II HI H

—m—

|h(t)4 (t)

o— (b)

T

-IT

I I

T

, Imn.^fjl

HlTm-,---- .

-»m—

Ixlt)

(0

2T

o-- (d)

_-^/\/V

|x(f)|

2T

-o 2

th(t)A (t)x(t)

f

r[H(f).A (f).X(f)|

o o

Jttlll-xin.-N H

,

<e(

-J_2T 2T

4,(0 A,(ft

111 I

To

l«)

IMMH

i

[h(t)Ao(0x(t}]*A t(t)

i.

Trrrrr-.-]N -H

o(9)

Ihi'll

l[H(f).A (f).X(f)!A,(f|

\ Aliiii

N -I |

Fig. 21

64

are not at a frequency higher than the Nyquist rate there

will be an aliasing effect that will distort the waveform.

But since in most systems, and especially in any real time

radar system, the waveform must be processed in a finite time,

there are for computation only a finite set of samples. This

is equivalent to multiplying h(t)A (t) (Fig. 21 c) by a

rectangular function (Fig. 21 d) . Again, the previous spec-

trum must be convolved with the spectrum of the rectangle to

get the result in Fig. 21 (e) . As expected, the larger the

T , the smaller is the distortion in the spectrum. But aso

'

L

stated before, the output must be in a digital format. So it

is necessary to multiply in the frequency domain |H(f) * A Q (f)

* x(f)|, by a string of delta functions, separated by 1/TQ .

That is equivalent to the convolution in time of a string of

impulses separated by T with h(t)A (t) x(t). The final result

is in Fig. 17 (g) . As can be seen, the output of the DFT

processor is not a sampled replica of |H(f)|. Aliasing effects

and finite time of processing are responsible for the amount

of distortion introduced. In order to get the mathematical

expression for the DFT, it is only necessary to make a parallel

derivation to that of the graphical derivation already dis-

cussed. The time domain expression for the output (Fig. 17 g)

will be obtained; then the Fourier coefficients of that

periodic function represent the DFT.

Let h(t) be the input function. The operation of sampling

h(t) is equivalent to a multiplication by

65

+ 00

A ft) = E 6(t - KT)° K=-oo

T 00

so h(t)A (t) = E h (KT) 6 (t - KT)° K=-«

Since x(t) < t < T - T/2

otherwise

is to be multiplied by h(t)A (t)

N-lh(t)A (t)x(t) = E h(KT) 6 (t - KT)

K=0

where NT = To

then we have to sample the Fourier transform of h(t)A (t)x(t)

by a string of impulses in the frequency domain

M f>

= ?~ 6(f-

J/V-1 +oo

which infers A^t) =F [A^f)] = TQ T J_m &(t - r T

Q )

must be convolved with h(t)A (t)x(t) to obtain the time domain

answer

.

hD(t) [h(t)A ft)x(t)] * A-,(t) =

N-l + «>

= [ E h(KT)6(t - KT)] * [TrL ra

6(t-rT )]

K=0

+oo N-l= T

n r ^ «>[ E h(KT)6(t - KT- rT )]° " K=0

Since hn (t) is a periodic function with a period NT, in

order to avoid time aliasing x(t) is chosen in such a way that

the end points of x(t) don't coincide with sampling points.

66

If they did coincide, it would generate an additive effect

at the boundary. But NT is still equal to T . Also, the

Fourier transform of a periodic function is given by the

Fourier coefficients of a series expansion. So

Hn (nf ) = H n (~) = E+ °°

c 5(f-nf )D o J D i ~°° n o

where, T -T/21 , O '

c = Fp— / hn (t) exp - j—f— dt , n = , 1 , . .

no -T/2

U L

o

So , cn

, T -T/21

ro T

T .

o -T/2

oo n-1 27mt£ ^ h(KT)6(t - KT - rT ) exp -jA^r

r=-°°K=0 ° 1dt

Since the integration is only over one period TQ

= r

T -T/2

n -T/2

N-1 ? t

'

E S'(t - KT) exp -j—^K>0 °

dt

N-1Z h(KT) /

T -T/2

n K=0 T/2exp-j-

2~ 6(t - KT) dt

VurrT^ • 27rKnTE h(KT) exp -j —~

—

K=0 lo

Since T - NT

E h(KT) exp -j ^^ , n = 0, 1, 2,.n K=0

N

and the Fourier transform of hD(t) is

K— uN NT-

67

nTo see that HD (j^) is also periodic, it is necessary to

show that c^ is periodic with a period NT.

Let n = r + N

since exp -j2frK(r + N)

Nexp -j

2 7rKrN

which implies c (r + N) = c (r)i n J n K J

which implies HD (^) = H

D (^)

so there are only N distinct values that can be evaluated

Normally the function

nN-l

H (w) = Z h (KT) exp -j 2TrnK/N , n - 0, 1... N-l1N x

K=

is called the discrete Fourier transform (DFT) which relates

N samples in the time domain to N samples in the frequency

domain. The inverse discrete Fourier transform, is, as expected

iN_1

i vh (KT) = £ i H (R

n

f) exp j4jM

, K = 0, 1... N-lK-

So it is seen that with very simple complex multiplications

and additions there is conversion from one domain to the other.

The Fast Fourier transform is an algorithm to implement

the discrete Fourier transform which, by reducing the number

of multiplications reduces the processor time to perform the

algorithm. For a better understanding of the FFT, it must be

reduced to a matrix form.

Consider the DFT

N-l 9 vx(n) =

T. x (K) exp -j , n 0,

K=0 ° NN-l

68

if W = exp -j 2tt/N, then in a compact fo

x(n) = WnK

x (K)

rra

where x (n) x(o)

x(N-l)

> x (K) = x (o)

x (N-l)

WnK

Wv

Wv

W(N-1)(N-1)

the examination o£ the matrix equation shows that there will2

be N possible complex multiplications and N(N-l) complex

additions. If N = 2Y , and the FFT algorithm [12] is used, the

number of multiplications will be reduced to Ny/2 and the

number of additions to Ny . Figure 22 gives a comparison

between the FFT algorithm and the direct computation with

respect to the number of required multiplications.

As it was shown, complex notation is greatly used in the

mathematical structure of signal processing. So the need to

represent signals in complex notation becomes a necessity even

in their implementation. It will be seen how a sine wave type

of function is converted in order to be represented by a

complex number.

In Fig. 23, let the input signal be

S(t) = A sin [0Ip

± ud)t +

<fr ]

where w TT. and go, can be thought as the intermediate frequencyIF d

and doppler frequency in a radar receiver. If the local

09

ooo

toZo

<o0-

_l

Z>

5u.

O(Cuio

64 128 256 512

N (number of sample points)

1024

Fig. 22

70

INPUTSIGNAL

Fig. 23

71

oscillator (I/Q) generates 2 cos ooTp

and. -2 sin to t which

can be obtained with a 90° phase shifter, the in phase channel

(I) and the out of phase channel (Q) outputs are of the form

I = A cos (to -jt +cf) )

Q = A sin Odt + <|>

o )

Since the low pass filter retains only the difference frequency

The I and Q signals can then be thought as the real and imagin-

ary part of the complex signal

Z = I + j Q

If the I and the Q channels are sampled and converted to

sequences of numbers x„ and y„, the resultant sampled pair

can be considered a complex digital word.

AK

= XK

+^ y K

=l

AK'

eXp^ *K

where x„ = A cos (to , K T + <j> )

y = A cos (to , KT + A)' n d H o

|AK |

= A, cDK

= todKT + ^

assuming a sampling frequency f = ^. Since t^ varies linearly

with K, the complex digitized signal can be interjected as a

vector of amplitude A that shifts at every sampling time

KT, by todKAT.

72

F. SYNTHETIC APERTURE RADAR

Using the pulse compression techniques already mentioned,

it is possible to achieve very small values of range resolu-

tion. In many radar applications such as aerial photography

not only a very small range resolution is necessary but also

an equivalent small aximuth resulution is desired. The con-

ventional method of reducing azimuth resolution by very small

beamwidths will always require big antennas which, especially

in airborne radars, is a problem since space is a limiting

factor. Also, the fact that azimuth resulution is an increas-

ing function with distance makes it many orders of magnitude

higher than the range resolution even for small distances.

Consider a radar without a pulse compression processor.

Then the range resolution 6R is

6R =j p c = speed of propagation

B = bandwidthw

If the antenna has an aperture D, the beamwidth is 6g = -p- ,

So the azimuth resolution 6.- is

x Dn _ RA R = distance to the target6AZ

= RGB " T

if B =10 MHZ, f = 5000 MHZ, D = 10 yds and R = 100 mi,w

then SR = 16.4 yds

6 = 1300 yds

Synthetic aperture radar is a way of processing radar

signals in order to get an equivalent antenna aperture much

bigger than the real antenna to reduce 6AZ>

In the synthetic

73

aperture case a single antenna is translate i^ Liansiated along a lineThe received signals are then stored, and processed afte/the radiating element has travelled &^ ^^ ^synthetic aperture concent ran ho •oncept can be viewed through two differentaspects. From a doppler viewpoint or from » tF or trom a linear array view-point, the basic principle nf tv,„ ar-ixiiLipxe or tne dopnler P ffPr t „•^-ttI crrect viewpoint isthat there exists a one to one eorrespondence between the!ong track coordinates of a reacting object and the instan-taneous doppler shift.

Consider Fig. 24 in wMch a„ airplane ^^ ^ ^ ^.^h wxth a constant speed v and with a radar antenna thatilluminates the area A. The ra^r n n + Qme radar antenna does not rotatebut simply moves with the airplane.

If there is no pulse compression and the width of thetransmitted pulse iq T i + a c iPUise is T

,lt is known that the slant resolution

constant is

6'R 2

s2~T

w

so the corresponding ground resolution constant is

PR = SR sec

if,= £L sec

<(,

Also, from Fig. 24 it is seen that L . BR where B is the bwidth of the transmitting antenna and Le£f is the the alongtrack resolution p .

x

earn

SinceD

thenpx

= Leff

="d-

R

74

Ground range

Fig. 24

aircraft v x

reflecting

object

Fig. 25

75

and as pointed out, it is impossible to increase D beyond

certain limits in order to decrease p . But if a relation

is found between doppler frequency and the position x of the

radar, then it is possible to make an azimuth discrimination.

From Fig. 25 it is seen that the radial velocity with

respect to the target is

v = v sin 6

So the doppler shift will be f , = ~ f sin where f is thea c o o

transmitted frequency. But if 6 is small, which is the normal

physical situation since the total illuminated area is small

and R is big, then sin 8 - 6 = x -x, which makes

± , - T-=r (X " Xjd AR o '

So it is seen that with a frequency analysis of the received

signal, it is possible to get an azimuth resolution that is

only a function of discriminating between the closeness of

two frequencies. The basic principle for the explanation of

synthetic aperture radar using the linear array theory is the

following

:

In a linear array various elements are fed at the same

time and the received echo is also received at the same time,

but due to path differences a receiving pattern is generated.

In the synthetic aperture case there is only one transmitting

element that moves with constant speed along a line. So a

long antenna will be formed not by physical means but by signal

processing. After the radiating element has travelled a dis-

tance L r r- corresponding to the illumination time of the realef f 1 b

76

beam, the stored signals after weighting and phase shifting

resemble the signals received by a big linear array.

Two cases are important to differentiate: the unfocused

synthetic aperture and the focused synthetic aperture.

In the focused synthetic aperture radar if e = (Fig. 26 a)

by signal processing (for R f «) or by real situation (R = °°)

the signals all arrive with the same phase. In a real linear

array the angle selectivity is provided only during the recep-

tion of the signals. But in the synthetic case, since only

one element is radiating and is moving, it is necessary to

account for the phase shift due to the transmitted and received

path. So, due to the fact that all the round-trip phase counts

for the formation of the receiving pattern, as opposed to the

real case in which only the received path is responsible for

the receiving pattern, the effective beamwidth is

3 x.c - T~i instead of

"

ef£ 2 LeffL eff

Since the length of the synthetic aperture radar is equal

to the distance corresponding to the time of illumination,

L = ^r , where D is the horizontal aperture of the physical

antenna and R the distance to the target. (Fig. 24), then the

azimuth resolution is

6AZ=

3 effR =

2 Le££

R =2

The azimuth resolution is independent of X and R in the

focused case which is of course an important result. This

means that the smaller the antenna the higher the azimuth

resolution.77

f flight path

(a) Lit

(b)

target

Total voltage V=ZV

\T=iivi

(c)

Fig. 26

78

In the unfocused case it is important to account for the

phase difference between the center and the ends of the equiv-

alent linear array. A quantitative interpretation of the

unfocused synthetic aperture radar can be explained as follows

Consider Fig. 26 (a): Due to the fact that R / R, thereo '

is a phase shift between the center and the end. The total

voltage is the vector sum of small discrete voltages that are

received along the path Lff

(Fig. 26 b) . If the number of

discrete voltages is large, it is possible to make the approx-

imation of Fig. 26 (c) and determine the relationship between

the resultant voltage V and the linear sum of the discrete

voltages V . If the 3 db point is defined as the boundary

for the drop in voltage, a must be approximately tt/2. But,

when a - tt/2, the radius r, (Fig. 26 c) is perpendicular to

the tangent t-. , implying that the phase difference between

the voltage at the end and at the center must approximate 45°.

This sets a limit on L rr . So when R = R + A/8 (round tripeff o v

< > A/4) the conditions are met. Using Fig. 26 (a):

(Lef£ /2)

2 + R 2 = (R + A/8) 2 = R 2 + ^ + |i

\ 2

if ^-j is very small compared to the other terms L££

= / RA

Since 7J1 = .455 ,m -~ / A~

3db Le£f

2 -T

the azimuth resolution would be 6^ - R0 - j /X R

Since 6 A7 is a function of R ', this is a solution between

the focused case and the conventional case. Figure 27 gives

a relation between the three cases.

79

'0 looRANGE (nmi)

Fig. 27

80

The signal processing theory for synthetic aperture radar

can be understood through the analysis of the ambiguity func-

tion of the received signal. If it is possible to separate

the range and azimuth as the two ambiguity factors from the

general ambiguity function, then the determination of the

azimuth ambiguity function and consequently the azimuth reso-

lution function are immediate.

Consider a transmitted signal f(t). After reflection in

the ground or target with reflectivity p(x,y,z), the received

signal will be

S(t) = /// p(x,y,z) f (t - ^) dxdydt [13]

where the integration is made over the illuminated region,

and R is the distance between (x,y,z) and the radar antenna.

The processing of the signals in fact recovers a signal pro-

portional to p(x,y,z). Using matched filter techniques, the

output will be of the form,

eQ(t) = / f*(t - -~4s(t) dt

= //// P (x,y,z) f (t-^) f* (t -^-) dtdxdydz

where R' is the distance from the antenna to the point

(x',y',z') corresponding to ptx'jy'jZ 1

). Let's find the

ambiguity function of f(t). By definition it will be

<Kx,y,z,x',y',z') = / £(t - ^) f* (t - 2|1) dt

where * means complex conjugate.

81

So

eQ(t) = fffftyp (x,y,z) dxdydz,

is a weighted average of p(x,y,z) where the weighting function

is i^(x,y,z,x' ,y' ,z') .

Let f(t) = g(t) exp j uiot

theni r r+ 2R ^ * r + 2R'. . ,2R 2R\

, +* = f g(t - —) g* (t - —) exp 3 w (— - —) dt

Assume the energy is transmitted in finite amounts of

time. If during those finite lengths of time the exponential

does not vary too much, then during a transmission the expon-

ential part can be considered constant although varying between

transmissions. So it can be taken out of the integral. The

result will be

v r r f* 2^ * r+ 2R' ,. , . r 2R 2R'ij>

= Z [ / g:(t - —) g* (t —) dt] exp -j u)o(- —

)

If gg* is constant during a transmission period, it can be

moved out of the summation and

i> = [/gg*dt] 2 exp -j^ [-— - —

]

It is obvious that the first factor is the range ambiguity

function, and the second, the azimuth ambiguity function.

So, the azimuth ambiguity function is

r2R 2R'

ij,

AZ= Z exp -3 u)

Q(- —

)

Figure 28 indicates a real situation. From that figure

it can be seen that:

82

Fig. 2,

X = vt

R = /R < + 17 Ro

+2TT

o

Ri = /R- +(X - X')

I ^ 2

R + U - X')o 2R

o

if x << Rq

and (x 2- x') << R

q, which is basically the same

restriction given to 6 in Fig. 25.

If the radar transmits with a PRR of f D = 1/T, then x =K

nvT are the positions where transmission occurs.

which infers \\> .= z exp -j w (— - -^j

AZ r J o v c c

= Z exp 2^ (2xx- -(x')2

)

o

N/2= exp + j ^

—

J— z exp -j 4tt(x'AR ) nvT-N/2 °c R.

where L, the synthetic aperture length is

L = NvT , v = aircraft speed

which infers t|/.7

pXn i ^o( x ') 2 sin (N + 1 )4ttx'vT/2ARqp J Re sin 4ttx*vT/2AR

o ' o

fx ' ")2

where exp j~-^ is only a phase factor. TheRo

c

sin (N + l)a/sin a is the amplitude factor of interest. The

azimuth resolution is given by the 3 db points separation.

IfN

N^-r- - 1, the 3 db point is at

2ttx' L _ , .

AR„

and s = 2x , = i^ARaAZ ttL

84

Since L = 3RQ

= ^ RQ

6az=~t~

D K2

which is a result in agreement with that already derived.

In generating the synthetic aperture radar, in order to

reduce the side lobes of ^ A7 , it is sometimes useful to weight

the returned signals before combining them. In the case of

the focused antenna, phase compensation must also be used.

So, if N returns are processed in order to get

L = N v T

the focused processing will be of the form

I S [exp j \b ]Wn L ^ J y n J n

and the unfocused processing of the form

Z S Wn n

where W is the weighting function and S^ are the discreten n

returns. This is the basic processing technique. Various

methods, from optical, electronic, as well as acoustic, are

used to generate the synthetic aperture radar.

V. RECENT DEVELOPMENTS IN TWO MAJOR AREAS

A. DIGITAL MTI

The structure of an MTI processor clearly points into the

direction of a digital implementation. The increase in digital

data rates associated with the cost decay of present digital

components led to the implementation of reliable, flexible,

low cost digital MTI processors.

Fig. 29 is a typical block diagram of one of the types of

MTI digital implementation. I and Q channels are used in

order not to lose 3 db on the average due to blind phases.

The structure is parallel to the analog MTI processor. The

use of A/D's and D/A's, storage devices instead of delay lines,

make up the differences with respect to the analog processor.

The capacity of the memory depends basically on the number of

range cells and on the use of multiple cancellers.

The quantization process introduces a new dimension to the

noise problem: that is, the quantization noise. Quantization

noise is present along all the dynamic range of the processor,

but the errors at the extremes are of particular importance

to the MTI processor. At the lower level, clutter cannot be

cancelled below one bit of quantization; at the upper level,

abrupt clipping distorts the signal creating additional noise.

This extra noise affects the improvement factor (I) of the

processor. If it is assumed that signal and quantization

noise are uncorrelated , then for a single canceller, the

improvement factor is [14],

86

REC.SIGNAL

SAMPLEANDHOLD

" BITS RANGE gate

A DCOnv.

_ \

LOCAL j

osc. I

DIGITALSTORE On

l' + 0'

Fig. 29

87

Il1 + (opVE 2

) [P,(T) _es. (T)l - p (T) .+ Ca77Pi^Tl~T~pTn7T '

where

p£

= correlation coefficient of quantization error forclutter alone

p es= correlation coefficient of quantization error when

both clutter and signal are present

a2 - noise power

E = signal voltage

P. = input power clutter

p(T) = correlation coefficient of clutter signals.

Since o2 /E 2 = and p (T) =e e

1 - P(T) + a^/P i(

which differs from the equivalent analog factor only by the

term 2/P-j^.

Also, for a double canceller, the improvement factor I

is [14]

1I

1 " j P(T) + ~ p(2T) + a£

2/P ic

For the triple and higher order cancellers, the expressions

get more complicated but the derivation is the same for all

cases. Also in I , the only difference from the analog

expression is the factor a2/P- . Assuming a uniform errorrE IC

distribution

a 2 =F 2

,n-G 12(2il_

- 1/2) 2

..here Em = saturation voltage of the digital register

n = number of bits used in the quantization.

Pic /

ae

2is called the improvement factor limitation.

Figure 30 relates the improvement factor limitation with the

total number of bits for different values of P. /E 2.

ic m

In many applications the fact that the MTI processor has

blind speeds at multiples of the PRF is completely undesirable.

The method usually used to attenuate this effect makes use of

staggered PRF. The scan- to- scan staggered PRF is less efficient

but requires less complex hardware than the pulse- to- pulse

stagger, but for some applications the first one is inadequate.

The staggered PRF technique is based on the use of different

spacing between transmitted pulses which, at the receiver, are

properly delayed (de- staggered) , such that the spacings at

the input of the delay line cancellers are all equal. The

necessary use of delays makes staggered PRF perfectly matched

for digital techniques. Since the concept is completely

defined, the recent trend has been focused in the development

of optimization techniques in order to improve the performance

of the processor. The first difficulty with any optimization

problem is, of course, the definition of optimum. Some try

to optimize the improvement factor, others to optimize the

signal to clutter gain (SCG) within a finite number of fre-

quency slots, or even the optimization of some defined indicator

An important factor in the optimization process is the

statistical model that is chosen for the clutter power dis-

tribution. It has been verified that a Gaussian distribution

centered at zero doppler frequency is adequate in most cases

as a power density function of clutter. Various standard

89

deviations have been calculated for different types of

clutter [2]

.

Figure 31 is a block diagram of a staggered PRF processor.

The pulse staggered sequence is such that the i^' 1 pulse is

delayed (i - 1)T + AT. seconds with respect to the first pulse.

In the processor, after de- staggering , the pulses are weighted

(W.'s in Fig. 31). By observation, the impulse response of

the processor is

h(t) = W1

6(t - AT.,) + W2

6[t - (T + AT2 )]

+ ... + WN

6{t - [(N-l)T + ATN ]}

which implies a transfer function

H(u>) =i l 1

\ij

exp {-j u [(i - 1)T + AT.]}

and a filter power response

G(co) = H(oo) H*(uO , * = complex conjugate.

With G(w) and the power spectral density of the clutter,

different optimization criteria can now be devised. One factor

is common to all techniques. The A-'s, W.'s and the cluttern i ' i

variance, assuming a Gaussian distribution, are the parameters

that will optimize the processor, given an optimization pro-

cedure. In the analysis, it is always assumed a step scanned

array antenna in order that for each position, N pulses are

transmitted. Figure 31 is a processor for a given range cell.

If there are K large cells, the general block diagram is

presented on Fig. 32, where W.'s and the summing network is

presented for the K. cell.^ i

90

5 6 7TOTAL NUMBER OF BITS

Fig. 30

90cl

Pulte Stagger Sequence

AT, T+iT (t-l)TtriT• •

i

lint

Processor Configuration

(N-l)HtiTM

tn-tATj

Input—4 Delay « cTj

[

Weight W|

Delay . T + &T-,| weight I,

I u-| Delay"- (X-DT - 6TN fc—^/W

tput

Fig. 31

TKC

/;, ,'ec/«3C n

•^C

I----JLJ

o-

m 'N-lo

-cOutput

Fig. 32

91

The ideal MTI filter is a high-pass filter with rejection

from zero frequency to the maximum frequency of the clutter.

Rejection may also occur after the maximum desired doppler

frequency but it is not absolutely necessary.

For a better understanding of staggered PRF some general

concepts should be explored. If a constant PRF is used, not

only the dc component of the doppler spectrum but the whole

clutter spectrum is translated to multiples of the PRF. It

is shown [15] that with staggered PRF the clutter spectrum

still is translated to multiples of the average PRF, but the

spectrum of the signal is dispersed into N (number of A-'s)

separate frequency lines. So, if a target produces a doppler

frequency that is a multiple of the basic PRF, only one part

in the whole signal is distorted by the clutter spectrum

where the remaining M-l parts of the signal may possess enough

power to be detected. An MTI filter which notches in a region

around dc as well as in multiples of PRF can eliminate the

clutter, but if staggered PRF is used only one part in N of

the signal is lost, thus increasing the probability of detec-

tion of targets with potential blind velocities.

In [16] an optimization process, based on a performance

index P defined as

Bu 1P = /» u df

B l [S*(f)]M

is developed. B-. and B are the lower and upper bounds of the

velocity region, S 2 (f) is the signal to clutter gain and M is

a parameter that reflects the emphasis the criteria puts on

92

spectral amplitude variations. As M increases, smaller

spectral amplitudes have more influence in the final result

of the integral.

For an understanding of the significance of P, it should

be noted that good clutter attenuation as well as a high im-

provement factor correspond to a good performance index.

Using the optimization procedure described in [16] and using

M = 8, practical results (figures 32, 33, 34) for different

values of N and (B - B,) were obtained. It can be seen that

as N increases, the minimum value for the signal-to-clutter

gain in the velocity region increases as well as the improve-

ment factor I = SCR(f) (Fig. 32, 33), but with a fixed N and

the velocity region increased on the upper bound, the minimum

value of SCR(f) in the velocity region decreases (Fig. 33, 34)

As was said earlier, the clutter variance a is also an

important parameter in the optimization process. Fig. 35

relates a with the improvement factor for the case of N = 5

and B = 10 B, . In the solid line the interpulse periods

T + A- were optimized for the value of o = 0.02 B, and the

coefficients W. were optimized in parallel with the variation

of a in the horizontal axis. On the dotted line, both coef-

ficients and interpulses were optimized for a = 0.02 B-, , and

the variations of I with a were recorded. This situation

occurs when the real clutter parameters are not those used

in the design of the filter. It is apparent that there are

not too many differences between the two cases.

93

5 6 7 9 10

SCR gain as a function of Doppler frequency;N~4.BU - 1 0ff,, a =0.02/?, . Optimum intervals: 1.100.1.094, 1.000. Optimum coefficients: 1.000 —3 0033.155.-1.152.

Fig. 32

5 KO

imprrs>vT+nt factor :7t9d8

I 5 6 7 8 9 10

» Dcppler frequency ItBjHzl

SCR gain as a function of Doppler frequency;

a = 0.02S, . Optimum intervals: 1.111,

1.000, 1.091, 1.058. Optimum coefficients:

-4.258.6.242.-^.031. 1.048.

N = 5, fl,. = 10S

1.000.

Fig. 33

94

SCR gain as a function of Doppler frequency;

N — S.B ~ 4 Off, . a - 0.028 ,.Optimum intervals: 1.007.

1.021, 1.000, 1.209. Optimum coefficients: 1.000,

—3.784. 5.432. -3.273, 0.626.

Cj MX,5

_....

Sirrprovtrr&nt loc'V s 751 JB 1MM»'MW^^^f^f^A^A

S 60

li

40

liiiJi • - 5>6JS

H , 5

$, * iOB,

20« O-?0;

°<) » io x> u

» Doppler frequency 1 (B\Hz)

Fig. 34

Improvement factor as a function of clutter spectral width

o;N ~B, Bu = 1 0ff, . Coefficients for dotted curve (optimum for a =0.02£,): 1.000, -4.258, 6.242, -4.031, 1.048. Intervals for both

tolid end dotted curves (optimum for o = 0.025,): 1.111, 1.000,

1.091,1.058 (c.f.. Fig. 8).

120

100

CO

co

K

\i 1 1

coefficients rr*?rcrvd to O*nre rpu-'$4 cc-ocs meters toG=G2Bi

y'

i! \toth eoe!t<*ntsor.d ir?&-pu.'se pe-'o

mclcrea to C» C? B

Ns.

•T^-*^

hi,

5

Bu'fOB,"*"*""" — »

XI? « £6 JD6

-*• IB,Url

07 JDS

Fig. 35

95

Another criterion to optimize the staggered PRF processor

is developed in [17]. Instead of maximizing the average sig-

nal-to-clutter gain over the entire velocity region, the

signal-to-clutter gain is maximized in small Af intervals

within the velocity region. The number of intervals is, of

course, directly related to the complexity of the processor.

Using the optimization algorithm developed in [19] , a final

block diagram for the processor is reached (Fig. 36, 37). In

Fig. 36 the input signal after being split in I and Q channels

is weighted with the optimum set of weights A and B for each

channel. The results are then combined in order to get an out-

put of positive doppler as well as an output of negative doppler,

Fig. 37 represents the implementation of the filter that gener-

ates the sets of A and B weights for each of the M frequency

sub- intervals in a channel. The T.'s are the interpulse delays

and the

, < i < N - 1a . . b . .

1J ' 1J < j < M - 1

are the coefficients determined by the optimization process.

It should be noted that in this optimization process the

interpulse periods are not optimized: only the weights are

optimized. Fig. 38 relates the signal-to-clutter gain aver-

aged over the entire velocity region with the signal-to-gain

maximized in each of eight frequency sub- intervals . Both

N = 5 and N = 10 are presented. A substantial improvement

due to optimization is evident. Another optimization tech-

nique is developed in [18].

96

HOOVIATIOHCOHPOHtHT

U-A"n£IGMTlNO »^K

I riLTia V'

( *

,,SAUPltA

<xnnjT

rxU- (I.! OAT rvt

* iAHfllR

Fig. 36

OUTM/'S O*M W! 'OMTIVS

fllTtts(t KtlCHTO

U TitlGnTiM3

UlTlflS

Fig. 37

97

The optimized processors are called Constrained Improvement

MTI Processors (CIP) . Given a specified improvement factor,

the number of pulses to be processed and the PRF stagger

sequence, the mean square deviation from a constant response

in the velocity region is minimized. Two main approaches can

be given to the problem. In the first, the PRF stagger

sequence is fixed and the weighting coefficients are chosen

in order to optimize the processor; in the second, the weight-

ing coefficients are fixed while the stagger sequence is chosen

in order to optimize the processor. Defining

F *

f = tt- , F' = maximum PRF

F = unstaggered PRF

figures 59, 40 and 41 represent the optimum filter response

with a four pulse return, f = 8, o - 0.01 and I = 30 db

,

using respectively a linear PRF stagger with ±201 interpulse

variation, and a sinosoidal PRF stagger with ±10% (Fig. 40)

and ±90% (Fig. 41) interpulse variation. As it can be seen,

as the variation increases the response becomes more uniform.

Comparing figures 42, 39 and 43, in which the number of pulses

are respectively 3, 4 and 6 for a common interpulse variation

of 20%, it can also be seen that the response improves with

the number of pulses processed.

A comparison between optimization procedures, different

from a direct comparison between responses, can be devised

using the fraction of frequencies for which a response is

less than some specified value as a comparison parameter.

If the db scale corresponds to the mean value of the

97a

r«lOuf*cv iM t nv*t— *«««MSt:c»iN«uxi<<tItB«i

NAL-TOCLUTTER GAINTWO TYPES OF N-PL'LSE

CELLEH. o - RMS CLUTTERRAL SPREAD; Tmin - MINIMUMSPACING. STAGGERED SPACCVC.MBER OF PULSES

Fig. 38

OOPPURFREoVENCrif/PRFJ 8'°°

Frequency response of a four-pu!

rpulse period variation"

«a99er with * 20 percent inmp™'"''"''8 C ' P U$,

'

nfl'inMr PRF

Fig. 39

98

Frequency response of four-pulse CIP using sinusoidal PRFStagger with ±10 percent interpulse period variation.

S

Si

'0.00 .00 J. 00 3.00 «-01 J 00 6-00 1.00 (.00DGPPlER FREQUENCY! r/PRF ]

Fig. 40

Oo

"o.oo i .oo 200n « n «

>;03 '- 00 5 °° «-oo

OOPPltP. FRE0UENCY(F/PRF|700 1.00

Frequency response of four-pulse CIP using sinusoidal PRFttaggcr with t 90 percent interpulse period variation.

Fig. 41

99

'O-OO 1.00 }.C0 SCO 4. CO 5.00 6.00

OOPPLER FRE2uEKCY[F/PRF)

Frequency response of three-pulse CIP with ± 20 percent

interpulse period variation.

Fig. 42

Response of six-pulse CIP using sinusoidal PRF stagger

with ± 20 percent variation in interpulse period.

£

Voo I .00 3 00 4.03 5.00 tOOOOPPLER fREOUEKCtlf/PRF )

Fig. 43

100

frequencies of interest, that means that 50% of the frequen-

cies are above as well as below the db level. This leads

to a cumulative distribution function in which instead of a

fraction of frequencies, the term probability is used. Fig.

44 is a cumulative distribution of the responses of a four

pulse CIP and a four pulse processor using the Ref. [16]

optimization criteria. This comparison criteria gives a

better performance for the CIP processor since its curve is

closer to being the ideal step response at db than the other

one, but on the average they are similar.

\ Another type of digital MTI processor is called the matrix

MTI . In this case there is no parallel between the analog and

the matrix MTI processor. This processor (Fig. 45), in addi-

tion to the tremendous flexibility in the modification of the

shape of the response curve, has associated with it a very

simple clutter locking mechanism that shifts the response

curve whenever there is an average velocity associated with

clutter. To simplify the explanation of the processor in Fig.

45, only two levels of quantization are used. Basically the

processor is a coherent system that compares the phase differ-

ence between two successive returns and weights that difference

in phase according to the shape it is intended to give to the

response curve. Since d<f>= u, dt, by weighting d4> in effect

the doppler is being weighted.

The phase angle of a signal return is defined by

<J>= the"

19.

10.1

-20-00 -10.00 0.00RESPONSEIDB)

Fig. 44

102

5 D* o

or q:UJ ui>- K-

V- z

_i ou u

•Hfin

103

and the amplitude by (I2

+ Q2

) . Since in the case of

Fig. 45 there is only one quantization level for I and Q,

that means that the only possible phases for I are 0° and

180° and for Q, 90° and 270°, assuming a signal to noise ratio

much greater than 1. When the average phase difference is 0°,

it implies a stationary target, but if d<$>T

= 180° that indicates

an optimum velocity target. Since each vector return can be

located in all of the four positions ±1 ± jQ, there are 4x4

possible combinations for two consecutive returns. So, what

the compression matrix block in Fig. 45 does is to generate

signals proportional to the phase difference output d(}> T

between two consecutive vector returns, which in this partic-

ular case can take the values 0°, 90°, 180° and 270°. Each

of the four outputs has a corresponding weighting factor W

and the result is summed to generate the MTI output. Fig. 46

indicates the four types with the vector combinations that

generated them, as well as the weighting factors used in order

to approximate an MTI response curve. The final response curve

corresponds to a statistical average of d<t> when uniformly

random phases are introduced at the input. In order to have

a signal proportional to the clutter velocity, it is only

necessary to subtract the outputs of the region (d) (Fig. 46)

from those of region (c) and divide by the number of range

gates, that is, for two quantization levels and a uniformly

distributed phase input

M = [(d) - (c)](90°)N

and VAR d<{> = — (45°)3N

104

CLUTTER -

CLUTTER OR *4*

OUTBOUND TARGET

TARGET-CLUTTER OR

INBOUND TARGET

-ISO-

MATRIX

ft)

(O

[iJq;

[l|Qt

t«T«T

[170*

i;q;i

ft)

(J)

l'i Qi

i.;o;

n;o; . i;o;i

;q;1

(b>

.-0-1

17051

MATRIX(c)

360° <J<jT

Fig. 46

105

So, in a large clutter environment if N is large, d<fZ can

smoothly adjust the local oscillator to place the MTI null

at the mean value of the clutter doppler.

Consider now an n quantization bit circuit. For each

channel there are 2 possible quantization levels. If only

one quadrant of possible phase differences is considered, it

can be seen that there are (Fig. 47)

2n

2n

2n— x — - — +1

2 2 2

possible phases. The -2" +1 parcel results from the fact

that of all the diagonal combinations (45° zone) only one

combination can be counted. As an example, let's use two

quantization bits. That means that there are

4[22*-2_

2n-l

+ 1} = 12

n = 2

possible phases in all four quadrants, which also implies 12

possible phase differences.

Figure 48 (a) represents the response and figure 48 (b)

is the matrix output (phase) for each 12 x 12 possible com-

bination of two consecutive vector returns. Since there is

flexibility in modifying the weighting coefficients, it is

possible to adapt the filter as a function of the type of

expected clutter spectrum. Again, the mean clutter velocity

can be determined by subtracting the outputs of the (b) region

(Fig. 48) from those of the (m) regions and applying the

result to the VCO associated with the local oscillator.

10G

m

2

Total number of combinations per quadrantcounting the diagonal combinations only asone =

n-l ~n-l ~n-l , ,

2 x 2 - 2 +1

Total number of combinations for the fourquadrants =

4[2n-l.

2n-l_

2n-l

+ i;

= 22n

2n+1

* 4

Fig. 47

107

i 2 3 4 6" 6 7 2 3 to u |2

rt?/

PULSE MO J PHASE SECTOR

1

1 2 } i s t 7 8 1 !0 tl 12

b < 4 • 1 s K ,j

k .

: - • b c 4 •

] I> „ • b c 4

pK < b bu1 >

•A1 1

1 kU< I h •1t-

7 *

oz 1 •M-J » •

10

II

11

•

•

Fig. 4

108

B. DIGITAL SAR

The synthetic aperture radar (SAR) is a typical example

of signal processing where digital techniques are being sub-

stituted for the former optical processors. With the present

technology mini- computers can compensate for the aircraft

movement, and, as with a real antenna, the synthetic antenna

can be steered and even scanned. The basic problem with SAR

processors is the need for high storage capabilities and very

fast data rates. The film as an optical storage device became

almost impractical since a real time display is impossible.

The use of storage tubes is also inadequate due to low effi-

ciency and poor dynamic range and stability. With the present

high speed A/D converters (100 MHZ), low cost, and high speed

compact digital storage devices, it becomes feasible for real

time SAR processors.

Table IV shows a time overview of the late achievements

in SAR techniques.

Figure 49 is a block diagram of a digital synthetic

aperture radar. Theoretically, the processor must, for each

range gate, perform the integral

/ s(t) r (t) d t

AT

where s(t) is the signal return and r(t) the correlator

reference function.

r(t) = AR(t) exp -j (f)

R(t)

AR(t) = weighting function to control the side lobes

of the synthetic antenna pattern.

^dC^) = phase reference that tracks the phase of s(t).

109

Date "* Development

1951 Carl-Wiley postulates doppler beamsharpening concept.

1952 University of Illinois demonstratesbeam sharpening concept.

1957 First SAR imagery using opticalcorrelator is produced.

Mid 1960's Analog electronic SAR correlationdemonstrated in non real time.

Late 1960's Digital electronic SAR correlationdemonstrated in non real time.

Early 1970' s Realtime digital SAR demonstratedwith motion compensation.

Table IV

110

tr*\

*-•

o

•HPh

111

Since <|> R(t) follows the phase of s(t) but with an unknown

difference, the signals are processed in an I (in phase) and

Q (90° out of phase) channel in order to prevent signal losses

The digital processor operates in the following way: The radar

returns pass an A/D converter that samples the signal at a

rate, at least higher than the Nyquist rate, and separates

the returns in range bins which are approximately equal to

the range resolution. The number of bits used per range bin

are a function of the desired dynamic range of the processor.

The digital data passes through a buffer and prefilter which

translates the A/D rate to the rate of the correlator. The

bulk memory stores the data corresponding to an integration

time, AT. The data is then correlated with the reference

generator output in order to form an azimuth line with a

length corresponding to the number of range gates. After

correlation, new data enters the memory and the process re-

peats itself to produce a new azimuth line.

From Fig. 50 it can be seen that the bulk memory in bits

must be equal to the number of cells times the average number

of bits per cell, so

BM = bulk memory = 2K N NDa k

NR

= number of range gates = —r

A TN = number of azimuth data lines = -™- = T f

The factor of 2 is because of the use of an 1 and Q channel.

But if, for example, the desired range resolution is 30 m,

that corresponds to a bandwidth

112

oLO

•H

113

BW = 2F" = ?x ^

8 m / gec = 5.0 MHZ ,a 2 x 30 m '

which implies an A/D sampling rate per channel of 5.0 MHZ.

On the other hand, for a 1 KHZ PRF the unambiguous range is

80 mi, which for a typical AR = 10 mi of mapping, means that

there is still an equivalent 70 mi excess time to process that

data. This results that in fact there is no need for the

correlator to work at such high data rates. Using a buffer

that accepts the data at a very high data rate during a short

period of time but delivers it at a lower rate to the corre-

lator, it is possible to reduce the correlator rates by many

orders of magnitude. In fact, analyzing the function of the

correlator, it can be seen that it must in AT seconds produce

a number of outputs equal to the total number of cells, that

is, (L/R )

N

D , where L/r = CA is the azimuth compression rate,a j\ a

Since the real antenna illuminates (L/r )Np

cells and is ready

to receive new data after T seconds, the correlator rate (CR)

is

CR =(L/r

a^NR = CA N D f

jR r

defining

N„ = K CA = total number of doppler filtersFa l l

K = azimuth over sample factor - 1

ThenCR = N c N D f

F R r

if Nr, = 500 and f = 1 KHZ for one doppler channel CR =R r ] F

.5 MHZ < A/D sampling rate.

114

On the other hand, if the spacing corresponding to V/T

is much finer than the range resolution r , that means excessive

data is heing used. If the PRF cannot be lowered due to power

or doppler considerations, then a prefilter can be used to

effectively reduce the PRf = f , by a factor f /f where fj r > ; s r r

is the prefilter sampling rate. The prefilter will reduce

the correlator rate by a factor f /f . If the required reduc-

tion is in order to have V/T of the order of r , thena

'

c V L 1 N F ...... . .

f ~ — ~ — —tf - —

?r > which implies a new correlator rate.s r r r T ' l

a a

CR = K K N^ N D fos s F R r

K = prefilter over sample factoros r r

k = synthetic array weighting

These two factors, the bulk memory and the correlator rate,

determine the type of design for a SAR processor. The main

objective is to reduce both CR and BM. Today's technology is

attacking this problem in two ways: devising correlator algor-

ithms which will reduce the bulk memory and arithmetic, at low

cost, with fewer power consuming elements. The first approach

is being made by parallel and series combinations of correlator

channels and prefilters, as well as with the use of FFT algor-

ithms. Another method is the use of pulse compression tech-

niques but processed only after the SAR processor. This will

reduce K which will reduce BM. The hardware improvements

are in CCD memories and LSI at much lower costs.

As it was said before, with today's digital techniques

various mapping modes are possible as well as motion compensation

115

Figure 51 shows the four types of mapping used today. The

Squint mode (Fig. 51 B) is only a variation in angle of the

side-looking SAR. The doppler beam sharpened mapping is a

PPI representation with an equivalent antenna of very high

azimuth resolution. The spotlight is a mapping where a high

resolution snapshot map is generated. Motion Compensation

includes various functions: clutter tracking, focusing accel-

eration compensation and real antenna stabilization. Fig. 52

is a block diagram of a general SAR processor. The blocks

are basically the same as those of Fig. 49 except for the

motion compensation blocks which are now included.

To achieve lower data rates, various types of algorithms

are presently used. Lower rates will be achieved at the

expense of more complex hardware. The general concept that

applies to all of them is the reduction of rates by the in-

crease in the number of doppler filters.

Figure 53 is a block diagram of the multi-channel prefilter

processor approach [19] where m is the number of channels.

The prefilter rate becomes

R - m ND fp R r

and since the number of filters Np

is reduced byl/m, the total

arithmetic rate (TAR) is

TAR = R +R = a, m + 2fc p 1 m

a = N n fi R r

a = K K N* N D /AT2 os s F R

, d TAR a2 nand —j = a - — = q

dm l m

116

ZF— ...

{AJSIDELOOK 3TWP MAPPING (B) SQUINT MODE STRIP MAPPING

/

'5'

A

(C) DOPPI.ER EEAM SHARPENED (D) Sl-OTUGHT MAPPINGMAPPING

Fig. 51

TRANS-

MITTER

STABLE

OSC.

\rOANTENNA

SERVO

RECEIVE!3(0

antenna command

AIRCRAFTMOTIONSENSOR

1REFERENCEFUNCTluM r(t)

COMPUTES*

/

SIGNAL PROCESSOR

CLUTTl" t

TRACKE ?

_____

clutter lock Servo error

Fig. 52

DISTLAY

117

CORRELATOR

FILTERS

PRE KILTER

SINCI.EPRE KILTER

Wm\Ymi\m\Y(mrr\

MULTIPLE

PREFILTERS

(A) FREQUENCY RESPONSE

PREFILTER CORRELATOR

1 I > ^Z.chzr.mU *

-ch2r^ie s ^

4(B) BLOCK DIAGRAM

Fig. 53

let STAGE KILTER3

2nd STAGE KILTERS

(A) FREQUENCY RESPONSE

1st STAGE Jnd STAGE

PREFILTS

yj-c ha tuie IB *

I 1 Pv 1

M »Tm -channel^ »

(B) ELOCK DIAGRAM

Fig. 54

118

implies that the value of m that minimizes TAR is

m = rQL2./jV2 = [K K N2 /f T

jl/2

lcx os s F r J

and the minimum TAR is

ot 1/ a 1/ 1/

TAR = a, (-*-) /2 + a (^) /2 = 2(a.aJ'2

mma

j2 a

2

= 2(K K f /AT) 1/z Nc NDv os s r ' F R

The two-stage correlator (Fig. 54) is another algorithm [19].

The idea is again frequency domain division but only with one

prefilter. The equivalent correlator rate will be the sum of

the two correlator rates. So

TAR = (KQs

Ks

NR

NF/AT) M

+ K K N D N C/MATos s R F

where M is the number of first-stage correlation channels.

1/As in the previous case, there is a value of M = (N

p )2 that

minimizes TAR. With the FFT algorithm, it is possible to

reduce even more the total rates. Table V is a summary of

the calculated values [19] of bulk memory as well as total

arithmetic rates for the SAR signal processing algorithms.

Another approach, completely different from those already

presented to solve the problem of high data rates on the

correlator (Fig. 49) of the SAR processor, is through the use

of parallel processing using associative memory [20]. Fig. 55

represents a block diagram of the associative memory and the

related input/output registers. In the associative memory,

119

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121

multi addressing, multi read/write, multi logical and arith-

metic operations are performed. The search/write and the

mask registers are used to generate proper codes for multi

accessing. The operations are done in a bit serial basis but

on all words in parallel at the same time. Ellis [20] showed

that for an X band radar with range and azimuth resolutions

of 10 meters, 1000 range cells, maximum range of 150 km and

maximum aircraft velocity of 200 m/sec, a 50 msec correlation

time is necessary: this implies a 40 nsec multiplication time

Using associative memory, it is possible [20] to perform the

operations of a correlation period in 31.48 msec. The advan-

tages of hardware parallel operations are thus obvious over

the conventional process when speed is an important factor.

122

VI. CONCLUSIONS. TRENDS

It was shown that due to improvements in integrated

circuitry, better microprocessors, faster DFT/FFT algorithms

and lower cost of digital logic, most analog processors are

being replaced by digital processors. Very high data rates

continue to be the major problem of digital processors, but

with today's techniques of parallel processing, general pur-

pose digital radar signal processors are already in use.

Despite the quantization noise inherent to digital processors,

good reliability, extreme flexibility and low cost of digital

processors still give them a tremendous advantage over the

former analog processors.

The future demand for more reliable and sophisticated

radar systems will be a function of cost and military neces-

sities. If the cost of digital logic trends lower as is its

present trend, and military demands continues high, digital

processors will play an increasingly important role in modern

radar system's implementation.

123

BIBLIOGRAPHY

1. Skolnik, Introduction to Radar Systems , Chapter 12,McGraw-Hill, 1962.

2. Shrader, Skolnik and others, Radar Handbook , pp. 17-18,McGraw-Hill, 3 970.

3. Oppenheim and Shafer, Digital Signal Processing , Prentice-Hall, 1975.

4. Rabiner, Gold, Theory and Applications of Digital SignalProcessing , Prentice-Hall, 1975

5. Berkowitz, Modern Radar, pp. 202-203, Wiley, 1965. "TIC&5"

6. Berkowitz, Modern Radar, p. 213, Wiley, 1965.

7. Berkowitz, Modern Radar, pp. 218-220, Wiley, 1965.

8. Nathason, Radar Design Principles, p. 522, McGraw-Hill, ,^\^

1969.

9. Nathason, Radar Design Principles, p. 329, McGraw-Hill,

1969.

10. Oppenheim and Shafer, Digital Signal Processing , Chapter 2,Prentice-Hail, 1975.

11. Oppenheim and Shafer. Digital Signal Processing , p. 206,Prentice-Hall , 1975.

'

12. Brigham, E., The Fast Fourier Transform , Prentice-Hall,1974.

13. Cutrona, Skolnik and others, Radar Handbook, pp. 23-9 to

25-15, McGraw-Hill, 1970.

14. Nathason, Radar Design Principles, pp. 564-565, McGraw-Hill,

1969.

15. McAulay, IEEE Trans AES-9 , No. 4, July 1973, p. 615.

16. Prinsen, I EEE Trans AES-9 , No. 5, September 1973, p. 714.

17. Urkowitz, IEEE 1975 International Radar Conference, p. 91.

18. Ewell, IEEE Trans AES-11 , No. 5, September 1975, p. 326.

19. Kirk, John, IEEE Trans AES-11, No. 3, May 1975, p. 326.

124

20. Ellis, IEEE Radar Present and Future Conf. No. 105,

pp. 311-317.

21. Gold and Rader, Digital Signal Processing , McGraw-Hill,1969.

125

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No. Copies

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Cameron StationAlexandria, Virginia 22314

2. Library, Code 0212 2

Naval Postgraduate SchoolMonterey, California 93940

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Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 95 94

4. Associate Professor John Bouldry 2

Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 95940

5. Lt. Joao P. Barcia 3

Calcada de S. Amaro, 27

Lisbon, Portugal

6. Curricular Officer 1

Electronics and Communications ProgramsNaval Postgraduate SchoolMonterey, California 939^0

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